We can adapt our approach to Matroid Median with Outliers in a relatively straightforward manner. Recall that our algorithm needs to start with an instance of outlier-free problem (i.e., Matroid Median) that provides a lower bound on the optimal cost of the given instance. To this end, given an instance \(\mathcal {I}= (X, F, \mathcal {M} = (F, \mathcal {S}), m)\)  of Matroid Median with Outliers, we define an instance \(\mathcal {I}^{\prime } = (X, F, \mathcal {M^{\prime }}, 0)\)  of Matroid Median with 0 Outliers (i.e., Matroid Median), where \(\mathcal {M}^{\prime } = (F \cup X, \mathcal {S}^{\prime })\)  is defined as follows. \(\mathcal {S}^{\prime } = \lbrace Y \cup C : Y \subseteq X \text{ with } |Y| \le m \text{ and } C \subseteq F \text{ with } C \in \mathcal {S}\rbrace \) . That is, each independent set of \(\mathcal {M}^{\prime }\)  is obtained by taking the union of an independent set of facilities from \(\mathcal {M}\) , and a subset of \(X\)  of size at most \(m\) . It is straightforward to show that \(\mathcal {M}^{\prime }\)  satisfies all three axioms mentioned above, and thus is a matroid over the ground set \(F \cup X\) . In particular, it is the direct sum of \(\mathcal {M}\)  and a uniform matroid \(\mathcal {M}_m\)  over \(X\)  of rank \(m\)  (i.e., where any subset of \(X\)  of size at most \(m\)  is independent). Note that using the oracle algorithm \(\mathcal {A}\) , we can simulate an oracle algorithm to test whether a candidate set \(C \subseteq F \cup X\)  is independent in \(\mathcal {M}^{\prime }\) . Therefore, using a \((2+\epsilon )\) -approximation for Matroid Median from [1]} in time FPT in \(k\)  and \(\epsilon \) , we can find a set \(A \subseteq F \cup X\)  of size at most \(k+m\)  that we can use to construct a coreset. The details about enumeration are similar to that for \(k\) -MedianOut, and are thus omitted.
